I'm very confused with separable extensions. I need to prove:
Let $E/F$ finite extension. Suppose there is an element $\alpha \in E$ which is not separable over $F$. Prove the existence of an element $E$ that is not in $F$ which is purely inseparable.
And I did the following: I think $\alpha$ isn't in $F$ (because if $\alpha \in F$ , then $F(\alpha)/F$ be a separable extension, and $\alpha$ isn't separable). Then I know $[E:F]=[E:F]_s [E:F]_i$, then $[E:F]_i=t= deg(g(x))$ with $g(x)=irr(\alpha, F)$. That is $g(x)= (x-\alpha)^{t}$ and I don't know how to follow.
But I think I'm wrong , I really need help.
It is as simple as that: Let $L$ be the intemediate field consisting of all purely inseparable elements. One can easily show that $E/L$ is separable, hence $L=F$ must be false.
Additional note (but you can do the proof without this): If we assume the extension to be normal - which we can do - $L$ is precisely the fixed field of $Aut(E/F)$.